In many high-frequency technology sectors (e.g., radio frequency integrated circuits (RFIC), high performance microprocessors, and advanced telecommunication packages), significant changes in wire impedance can be observed when compared to lower frequency applications. Increases in resistance (e.g., by a factor of two) and the emergence of frequency-dependent inductance effects that eventually become the dominant contribution to impedance are manifestations of the rising importance of time-dependent magnetic fields on the current distribution within conductors. The threshold beyond which wire impedance is desirably treated as a nontrivial computable function of frequency varies from technology sector to technology sector. This phenomena is better understood when expressed in terms of the skin depth δ:
                              δ          =                      1                                          π                ⁢                                                                  ⁢                μ                ⁢                                                                  ⁢                σ                ⁢                                                                  ⁢                f                                                    ,                            (        1        )            where σ is the conductivity inside the conductor, μ is the magnetic permittivity (usually that of the vacuum) and f is the frequency of operation. The skin depth is an approximate measure of the distance from the surface of a conductor that an external magnetic field can penetrate. As the frequency increases, the wire's minimum transverse dimensions become larger than δ, triggering the onset of frequency dependence in the electromagnetic parameters resistance R and inductance L. This happens at hundreds of MHz for packages, at a few GHz for passive on-chip devices, and in the neighborhood of 15 GHz for upper metal layer signals in ICs with 65 nm features sizes. For future process generations timing considerations, rather than scaling, will dictate the choice of critical wire transverse dimensions and with that, the resulting threshold frequencies.
In R. Escovar and R. Suaya, “Optimal design of clock trees for multi-gigahertz applications,” IEEE Transactions on CAD, vol. 23, March 2004, pp. 329.345; R. Escovar, S. Ortiz, and R. Suaya, “An improved long distance treatment for mutual inductance,” IEEE Trans. Computer-Aided Design, vol. 24, no. 5, pp. 783.793, May 2005; U.S. Patent Application Publication Nos. 2003/0131334, 2005/0120316, 2006/0282492, 2007/0225925, and 2007/0226659, all of which are hereby incorporated herein by reference, methods for extracting distributed frequency dependent electromagnetic parameters for wires in ICs are described. Some of the methods are based on correctly determining the loop impedance, and using sound physical considerations to identify what constitutes the return path. Once the return paths have been identified, the loop wire impedance is computed from solutions to Kirchhoff's current laws. The correct frequency behavior for inductance and resistance can then be reproduced, in terms of uniform current distribution for each wire. The solution to Kirchhoff's current laws is computationally efficient for frequencies below the skin depth, permitting full-chip dynamic impedance extraction at a reasonable computational cost up to about 15 GHz for 65 nm technologies. At higher frequencies, one can partition the wire into filaments to account for the nonuniformity of current distribution, resulting in an enlarged linear system whose order becomes mN with m being the number of filaments per wire and N being the number of wires. In certain embodiments, around 1000 wires/second can be processed for frequencies which are below the skin effect threshold. Above the skin effect threshold, it is desirable to set m ˜10 for a typical wire in order to achieve better than 1% accuracy in impedance extraction over a broad frequency domain (e.g., from about 15 Ghz up to about 100 Ghz). Using direct solvers, the computational performance drops to 1 wire per second, making the approach too expensive for all but selected critical wires. Accordingly, improved and computationally efficient methods for extracting electrical parameters for circuits operating at higher frequencies are desired.